共查询到20条相似文献,搜索用时 734 毫秒
1.
《Journal of Computational and Applied Mathematics》1998,88(1):57-69
For 1 ⩽k⩽n − 1 and 0 ⩽q⩽k − 1, solutions are obtained for the boundary value problem, (−1)n−k = f(x,y), y(i)=0, 0⩽i⩽k − 1, and y(i) = 0, q⩽j⩽n − k + q − 1, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone. 相似文献
2.
We solve the following problem. For 1 ⩽ j, k ⩽ n and |j − k| ⩽ m, let ajk be a given complex number with akj = ājk. We wish to find linearly independent vectors x1,…,xn such that 〈xk, xj〉 = ajk for |j − k| ⩽ m and such that the distance from xk to the linear span of x1,…,xk−1 is maximal for 2 ⩽ k ⩽ n. We construct and characterize all such sequences of vectors. Our solution leads naturally to the class of m-band sequences of vectors in an inner product space. We study these sequences and characterize their equivalence classes under unitary transformations. 相似文献
3.
《Journal of Combinatorial Theory, Series A》1986,43(1):130-132
An explicit formula is derived for the number of k-element subsets A of {1,2,…, n} such that no two elements in A are at “circular” distance q, i.e., if i ϵ A and 1 ⩽ i ⩽ n − q (resp. n − q + 1 ⩽ i ⩽ n) then i + q ∉ A (resp. i + q − n ∉ A). 相似文献
4.
《Discrete Mathematics》1986,62(2):113-118
The bipartite regulation number br(G) of a bipartite graph G with maximum degree d is the minimum number of vertices required to add to G to construct a d-regular bipartite supergraph of G. It is shown that if G is a connected m-by-n bipartite graph with m ⩽ n and n − m ⩾ d − 1, then br(G) = n − m. If. however, n − m ⩽ d − 2, the br(G) = n ⇔ m + 2l for some l satisfying 0 ⩽ l ⩽ d − (n − m). Conversely, if l, k and d (>2) are integers such that 0 ⩽ l ⩽ k and 2 ⩽ k ⩽ d, then there is an connected m-by-n bipartite graph G of maximum degree d for which br(G) = n − m + 2l, for some m and n with k = d − (n −m). 相似文献
5.
L.E. Trotter 《Discrete Mathematics》1975,12(4):373-388
We examine a family of graphs called webs. For integers n ? 2 and , the web W(n, k) has vertices Vn = {1, …, n} and edges {(i, j): j = i+k, …, i+n ? k, for i?Vn (sums mod n)}. A characterization is given for the vertex packing polyhedron of W(n, k) to contain a facet, none of whose projections is a facet for the lower dimensional vertex packing polyhedra of proper induced subgraphs of W(n, k). Simple necessary and sufficient conditions are given for W(n, k) to contain W(n′, k′) as an induced subgraph; these conditions are used to show that webs satisfy the Strong Perfect Graph Conjecture. Complements of webs are also studied and it is shown that if both a graph and its complement are webs, then the graph is either an odd hole or its complement. 相似文献
6.
A digraph G = (V, E) is primitive if, for some positive integer k, there is a u → v walk of length k for every pair u, v of vertices of V. The minimum such k is called the exponent of G, denoted exp(G). The exponent of a vertex u ∈ V, denoted exp(u), is the least integer k such that there is a u → v walk of length k for each v ∈ V. For a set X ⊆ V, exp(X) is the least integer k such that for each v ∈ V there is a X → v walk of length k, i.e., a u → v walk of length k for some u ∈ X. Let F(G, k) : = max{exp(X) : |X| = k} and F(n, k) : = max{F(G, k) : |V| = n}, where |X| and |V| denote the number of vertices in X and V, respectively. Recently, B. Liu and Q. Li proved F(n, k) = (n − k)(n − 1) + 1 for all 1 ≤ k ≤ n − 1. In this article, for each k, 1 ≤ k ≤ n − 1, we characterize the digraphs G such that F(G, k) = F(n, k), thereby answering a question of R. Brualdi and B. Liu. We also find some new upper bounds on the (ordinary) exponent of G in terms of the maximum outdegree of G, Δ+(G) = max{d+(u) : u ∈ V}, and thus obtain a new refinement of the Wielandt bound (n − 1)2 + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 215–225, 1998 相似文献
7.
《Journal of Number Theory》1987,25(3):308-312
If p(n, k) is the number of partitions of n into parts ≤k, then the sequence {p(k, k), p(k + 1, k),…} is periodic modulo a prime p. We find the minimum period Q = Q(k, p) of this sequence. More generally, we find the minimum period, modulo p, of {p(n; T)}n ≥ 0, the number of partitions of n whose parts all lie in a fixed finite set T of positive integers. We find the minimum period, modulo p, of {S(k, k), S(k + 1, k),…}, where these are the Stirling numbers of the second kind. Some related congruences are proved. The methods involve the use of cyclotomic polynomials over Zp[x]. 相似文献
8.
In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x1,x2,…,xk?nk there exist k distinct integers j1,j2,…,jk so that xi= aii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k). 相似文献
9.
The theory of vertex-disjoint cycles and 2-factors of graphs is the extension and generation of the well-known Hamiltonian cycles theory and it has important applications in network communication. In this paper we first prove the following result. Let G=(V 1,V 2;E) be a bipartite graph with |V 1|=|V 2|=n such that n≥2k+1, where k≥1 is an integer. If d(x)+d(y)≥?(4n+2k?1)/3? for each pair of nonadjacent vertices x and y of G with x∈V 1 and y∈V 2, then, for any k independent edges e 1,…,e k of G, G contains k vertex-disjoint quadrilaterals C 1,…,C k such that e i ∈E(C i ) for each i∈{1,…,k}. We further show that the degree condition above is sharp. If |V 1|=|V 2|=2k, we give degree conditions that G has a 2-factor with k vertex-disjoint quadrilaterals C 1,…,C k containing specified edges of G. 相似文献
10.
Jan Søreng 《Journal of Combinatorial Theory, Series A》1976,21(2):164-187
Ek(x2,…, xn) is defined by Ek(a2,…, an) = 1 if and only if ∑i=2nai = k. We determine the periods of sequences generated by the shift registers with the feedback functions x1 + Ek(x2,…, xn) and x1 + Ek(x2,…, xn) + Ek+1(x2,…, xn) over the field GF(2). 相似文献
11.
Jiuying Dong 《Journal of Applied Mathematics and Computing》2010,34(1-2):485-493
The theory of vertex-disjoint cycles and 2-factor of graphs has important applications in computer science and network communication. For a graph G, let σ 2(G):=min?{d(u)+d(v)|uv ? E(G),u≠v}. In the paper, the main results of this paper are as follows:
- Let k≥2 be an integer and G be a graph of order n≥3k, if σ 2(G)≥n+2k?2, then for any set of k distinct vertices v 1,…,v k , G has k vertex-disjoint cycles C 1,C 2,…,C k of length at most four such that v i ∈V(C i ) for all 1≤i≤k.
- Let k≥1 be an integer and G be a graph of order n≥3k, if σ 2(G)≥n+2k?2, then for any set of k distinct vertices v 1,…,v k , G has k vertex-disjoint cycles C 1,C 2,…,C k such that:
- v i ∈V(C i ) for all 1≤i≤k.
- V(C 1)∪???∪V(C k )=V(G), and
- |C i |≤4, 1≤i≤k?1.
12.
《Statistics & probability letters》1986,4(4):211-215
The probability distribution of the number of success runs of length k (⩾1) in n (⩾1) Bernoulli trials is obtained. It is noted that this distribution is a binomial distribution of order k, and several open problems pertaining to it are stated. Let Sn and Ln, respectively, denote the number of successes and the length of the longest success run in the n Bernoulli trials. A formula is derived for the probability P(Ln ⩽ k | Sn = r) (0 ⩽ k ⩽ r ⩽ n), which is alternative to those given by Burr and Cane (1961) and Gibbons (1971). Finally, the probability distribution of Xn, Ln(k) is established, where Xn, Ln(k) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k. 相似文献
13.
《Statistics & probability letters》1986,4(2):101-105
The probability distribution of the numbeer of success runs of length k ( >/ 1) in n ( ⩾ 1) Bernoulli trials is obtained. It is noted that this distribution is a binomial distribution of order k, and several open problems pertaining to it are stated. Let Sn and Ln, respectively, denote the number of successes and the length of the longest success run in the n Bernoulli trials. A formula is derived for the probability P(Ln ⩽ k | Sn = r) (0 ⩽ k ⩽ r ⩽ n), which is alternative to those given by Burr and Cane (1961) and Gibbons (1971). Finally, the probability distribution of Xn, Ln(k) is established, where Xn, Ln(k) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k. 相似文献
14.
A regular graph G = (V, E) is a k-stratified graph if V is partitioned into V1, V2, …, Vk subsets called strata. The stratification splits the degree dv ∀v ϵ V into k-integers dv1, dv2, …, dvk each one corresponding to a stratum. If dv1 = dv2 = … = dvk ∀v ϵ V then G is called regular uniform k-stratified, RUks(n, d) where n is the cardinality of the vertex set in each stratum and d is the degree of every vertex in each stratum. For every k, the class RUks(n, d) has a unique graph generator class RUls(n, d) derived by decomposition of graphs in RUks(n, d). We investigate the minimization of the cardinality of V, the colorability, vertex coloring and the diameter of the graphs in the class. We also deal with complexity questions concerning RUks(n, d). Some well-known computer network models such as barrel shifters and hypercubes are shown to belong in RUks(n, d). 相似文献
15.
《Discrete Mathematics》1986,58(1):63-78
In this paper we give a procedure by which Hamiltonian decompositions of the s-partite graph Kn,n,…,n, where (s − 1)n is even, can be constructed. For 2t⩽s, 1⩽a1⩽…⩽at⩽n, we find conditions which are necessary and sufficient for a decomposition of the edge-set of Ka1a2…,at into (s − 1)n/2 classes, each class consisting of disjoint paths, to be extendible to a Hamiltonian decomposition of the complete s-partite graph Krmn,n,…,n so that each of the classes forms part of a Hamiltonian cycle. 相似文献
16.
Let k≥1 be an integer and G=(V 1,V 2;E) a bipartite graph with |V 1|=|V 2|=n such that n≥2k+2. Our result is as follows: If $d(x)+d(y)\geq \lceil\frac{4n+k}{3}\rceil$ for any nonadjacent vertices x∈V 1 and y∈V 2, then for any k distinct vertices z 1,…,z k , G contains a 2-factor with k+1 cycles C 1,…,C k+1 such that z i ∈V(C i ) and l(C i )=4 for each i∈{1,…,k}. 相似文献
17.
《Applied Mathematics Letters》2001,14(3):347-352
We consider the following boundary value problem: −Δny = F(k,y, Δy,…,Δn−1y), k ϵ Z[n − 1, N], Δiy(0) = 0, 0 ≤ i ≤ n − 2, Δpy(N + n - p) = 0, where n ≥ 2 and p is a fixed integer satisfying 0 ≤ p ≤ n − 1. Using a fixed-point theorem for operators on a cone, we shall yield the existence of at least three positive solutions. 相似文献
18.
《Journal of Combinatorial Theory, Series A》1986,42(2):200-206
For integers n ⩾k⩾ 1 and L ⊃ {0, 1,…, k − 1};, m(n, k, L) denotes the maximum number of k-subsets of an n-set so that the size of the intersection of any two among them is in L. It is proven that for every rational number r ⩾ 1 there is a choice of k and L so that cnr < m(n, k, L) < dnr, where c, d depend on k and L but not on n. 相似文献
19.
Let t(k,n) denote the number of ways to tile a 1 × n rectangle with 1 × 2 rectangles (called dominoes). We show that for each fixed k the sequence tk=(t(k,0), t(k,1),…) satisfies a difference equation (linear, homogeneous, and with constant coefficients). Furthermore, a computational method is given for finding this difference equation together with the initial terms of the sequence. This gives rise to a new way to compute t(k,n) which differs completely with the known Pfaffian method. The generating function of tk is a rational function Fk, and Fk is given explicitly for k=1,…,8. We end with some conjectures concerning the form of Fk based on our computations. 相似文献
20.
Given graphs H1,…, Hk, let f(H1,…, Hk) be the minimum order of a graph G such that for each i, the induced copies of Hi in G cover V(G). We prove constructively that f(H1, H2) ≤ 2(n(H1) + n(H2) − 2); equality holds when H1 = H 2 = Kn. We prove that f(H1, K n) = n + 2√δ(H1)n + O(1) as n → ∞. We also determine f(K1, m −1, K n) exactly. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 180–190, 2000 相似文献